It has been shown that the SP equation (REF ) admits a Wadati-Konno-Ichikawa type Lax pair and is related to the sine-Gordon equation through a chain of transformations [1]}.
Then bi-Hamiltonian structure and the conservation laws were studied by Brunelli [2]}, [3]}. Sch\(\ddot{a}\) fer and Wayne proved the nonexistence theorem that the equation (REF ) doesn't possess any solution representing a smooth localized pulse moving with constant shape and speed [4]}. Sakovich and Sakovich also found the loop-soliton solutions of the SP equation (REF ) [5]}. Matsuno found the connection between the SP equation and the sine-Gordon equation through the hodograph transformation, and further
found many kinds of exact solutions including multi-soliton, multi-loop, multi-breather and Periodic solutions [6]}, [7]}.
And a lot of generalizations of the SP equation has been studied, for example, Pietrzyk, Kanatt\(\breve{s}\) ikov and Bandelow introduced the vector SP equation [8]}, a two-component SP equation that generalizes the scalar (REF ) and describes the propagation of polarized ultra-short light pulses in cubically nonlinear anisotropic optical fibers, which can be written as
\(u_{n,xt}=c_{ni}u_i+c_{nijk}\left( u_iu_ju_k\right) _{xx},\) 
